Projection of Submodules

Question

I have trouble working on this problem.

(1) What the definition of projection on $M$?

(2) Is $\pi_i$ endomorphism of $M$?

(3) To show (i), (ii), (iii), is that true if we show $\pi_i=1$ for only one $i$, and the other $\pi_i=0$?

Answer 1

Hint: Consider the projections $\pi_i:M\rightarrow M$ where $\pi_i(m_1,\ldots,m_i,\ldots,m_n) = m_i$. So the image of $\pi_i$ is exactly $M_i$.

(1) This property means that $\sum_i\pi_i$ is the identity map on $M$. (2) and (3) follows easily from the definition of projections.